Why do we 'invert and multiply' when dividing fractions?

Question

I read in a textbook that the explanation to this rule lies in the fact that division is the reverse operation to multiplication. Unfortunately, the author did not elaborate on this point. Based on this, can someone help me to understand why

$\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{a}{b}$ * $\frac{d}{c}$?

Answer 1

Let us say you have a number $X$ that you multiply by $Y$. Then you got $XY$. If you want to revert that operation, you have to divide by $Y$:

$$\frac{XY}{Y}$$

But that is the same as multiplying by $\frac1Y$:

$$\frac{XY}{Y} = XY \cdot \frac1Y$$

That means, like you pointed out, that division is the reverse of multiplication. But division by $Y$ is just multiplication by $\frac1Y$.

Let us say that both $X$ and $Y$ are fractions.

That is,

$$X = \frac{a}{b},\ Y = \frac{c}{d}$$

Let us also note that $\frac{1}{Y}$ now becomes $$\dfrac1{\frac{c}{d}} = \frac{d}{c}$$

Therefore, $$X / Y = X \cdot \frac1Y = \frac{a}{b}\cdot\frac{d}{c}$$

Answer 2

If you accept that you can multiply the numerator and denominator by the same nonzero number without changing the value of the fraction, then just multiply the numerator and denominator by $d/c$: $$ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b}\cdot \frac{d}{c}}{\frac{c}{d}\cdot \frac{d}{c}} = \frac{\frac{ad}{bc}}{1} = \frac{ad}{bc}. $$